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(a - b)/(a + b) is rational if a^3 + b^3 + c^3 - 3abc is rational

Source: 2015 VMEO IV Juniors 12.1 Vietnamese Mathematics e - Olympiad https://artofproblemsolving.com/community/c2463156_2015_vmeo_iv

September 15, 2021
rationalalgebra

Problem Statement

Given a set S⊂R+S \subset R^+, S≠∅S \ne \emptyset such that for all a,b,c∈Sa, b, c \in S (not necessarily distinct) then a3+b3+c3−3abca^3 + b^3 + c^3 - 3abc is rational number. Prove that for all a,b∈Sa, b \in S then a−ba+b\frac{a - b}{a + b} is also rational.
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